1. ## orbit

Suppose $f,g \in X$ and have the same pattern. Prove that $W(f) = W(g)$.

Now an orbit of and element $f$ in $X$ is all the "possible stuff" that a permutation $\pi$ can map $f$ to. So does $f,g \in X$ imply that they are in the same orbit?

2. Originally Posted by Sampras
Suppose $f,g \in X$ and have the same pattern. Prove that $W(f) = W(g)$.

Now an orbit of and element $f$ in $X$ is all the "possible stuff" that a permutation $\pi$ can map $f$ to. So does $f,g \in X$ imply that they are in the same orbit?
I'm sorry. Maybe some other member will swoop in and help you. But if not, I just have two questions. What does 'pattern' mean in this context? and what is $W(y)$ represent. Also, what is $X$? Just some group under consideration? If so, then $f,g\in X$ does not mean that $f\in\mathcal{O}(g)$ or vice versa (which is really the same thing).

3. Originally Posted by Sampras
Suppose $f,g \in X$ and have the same pattern. Prove that $W(f) = W(g)$.

Now an orbit of and element $f$ in $X$ is all the "possible stuff" that a permutation $\pi$ can map $f$ to. So does $f,g \in X$ imply that they are in the same orbit?

If you don't properly define your "stuff" people won't properly understand it: what are f,g? What is a "pattern" for them? What is W(f)? What is X?
What action of what group (presumably $S_n$) on what set are we talkin about here?

Tonio